Maximum Planar Subgraph on Graphs not Contractive to K5 or K3,3

The maximum planar subgraph problem is well studied. Recently, it has been shown that the maximum planar subgraph problem is NP-complete for cubic graphs. In this paper we prove shortly that the maximum planar subgraph problem remains NP-complete even for graphs without a minor isomorphic to K5 or K3,3 , respectively

A Note on the Practicality of Maximal Planar Subgraph Algorithms

Given a graph $G$, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of $G$ with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but---to the best of our knowledge---they have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, focusing on practical runtime, solution quality, and implementation complexity. Surprisingly, a seemingly only theoretically strong approximation forms the building block of the strongest choice.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems

In Michael Jünger and Petra Mutzel Algorithmica, 16 (1996) we used a branch-and-cut algorithm in order to determine a maximum weight planar subgraph of a given graph. One of the motivations was to produce a nice drawing of a given graph by drawing the found maximum planar subgraph, and then augmenting this drawing by the removed edges. Our experiments indicate that drawing algorithms for planar graphs which require 2- or 3-connectivity, resp. degree-constraints, in addition to planarity often give ''nicer'' results. Thus we are led to the following problems: 1. Find a maximum planar subgraph with maximum degree d in IN. 2. Augment a planar graph to a k-connected planar graph. 3. Find a maximum planar k-connected subgraph of a given k-connected graph. 4. Given a graph G, which is not necessarily planar and not necessarily k-connected, determine a new graph H by removing r edges and adding a edges such that the new graph H is planar, spanning, k-connected, each node v has degree at most D(v) and r+a is minimum. Problems (1), (2) and (3) have been discussed in the literature, we argue that a solution to the newly defined problem (4) is most useful for our goal. For all four problems we give a polyhedral formulation by defining different linear objective functions over the same polytope which is the intersection of the planar subgraph polytope, see Michael J{\"u}nger and Petra Mutzel Proc. IPCO3 (1993), the k-connected subgraph polytope M. Stoer LNCS 1531 (1992) and the degree-constrained subgraph polytope. We point out why we are confident that a branch-and-cut algorithm for the new problem will be an implementable and useful tool in automatic graph drawing

Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments

Given a graph G, the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of G with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski\u27s famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance

Maximum Planar Subgraphs and Nice Embeddings: Practical Layout Tools

In automatic graph drawing a given graph has to be layed-out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard. We attack the maximum planar subgraph problem with a branch-and-cut technique which gives us quite good and in many cases provably optimum solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is defined and studied. All subgraphs of a graph G, which are subdivisions of K5 or K3,3, turn out to define facets of this polytope. For cliques contained in G, the Euler inequalities turn out to be facet-defining for the planar subgraph polytope. Moreover we introduce the subdivision inequalities, V2k inequalities and flower inequalities all of which are facet-defining for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated. We also present computational experience with a branch-and-cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision of K5 or K3,3. These structures give us inequalities which are used as cutting planes. Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for finding nice embeddings by applying this method to graphs taken from the literature